Question: A circle is tangent to the lines $4x - 3y = 30$ and $4x - 3y = -10.$  The center of the circle lies on the line $2x + y = 0.$  Find the center of the circle.
Note that the lines $4x - 3y = 30$ and $4x - 3y = -10$ are parallel, so the center of the circle lies on the line which is exactly halfway between these lines, which is $4x - 3y = 10.$

[asy]
unitsize(2 cm);

pair A, B;

A = dir(-20);
B = dir(160);

draw(Circle((0,0),1));
draw((A + 1.5*dir(70))--(A - 1.5*dir(70)));
draw((B + 1.5*dir(70))--(B - 1.5*dir(70)));
draw((1.5*dir(70))--(-1.5*dir(70)),dashed);

label("$4x - 3y = -10$", B + 1.5*dir(70), N);
label("$4x - 3y = 30$", A + 1.5*dir(70), N);
label("$4x - 3y = 10$", -1.5*dir(70), S);

dot((0,0));
[/asy]

Solving the system $2x + y = 0$ and $4x - 3y = 10,$ we find $x = 1$ and $y = -2.$  Therefore, the center of the circle is $\boxed{(1,-2)}.$